In a wireless communication system, the characteristics of the propagation channel depend on the paths taken by a signal between transmit and receive antennas. These paths are functions of the stationary and moving reflector objects in the propagation environment, as well as the separation distance between the two antennas, which may also be changing. Furthermore, the presence of reflecting objects and scatterers in the propagation environment results in multiple replicas of the transmitted waveform that arrive at the receiver with different amplitude and phase, time delays and spatial orientations.
At a receiver, the incoming waves (multi-paths) arrive from different directions with different propagation delays, and with randomly distributed amplitudes, phases, and angles of arrival. When these multi-paths combine vectorially at the receiver, they cause the received signal to fade due to the constructive and destructive effects of multi-path wave combining at the receiver. Generally, the received signal is represented, mathematically, by convolving the transmitted signal with the channel impulse response, which can be modeled as a tapped delay line having a number of taps corresponding to the significant number of delayed receive signals.
In mobile wireless communications systems, combining of the multi-paths, with their random phases and amplitudes, results in rapid fluctuations in the signal's strength. It is possible to identify two types of fading, namely large-scale and small-scale fading. Large-scale fading represents the average attenuation in the signal power due to motion over large areas. Small-scale fading refers to the rapid fluctuation of the amplitude of a radio signal as a result of multi-path propagation over a short period of time, or equivalently, a short travel distance. Fading occurs as a result of the interference between two or more replicas of the transmitted waveform arriving at the receiver at two different time instants. When these signals combine at the receiver, the resultant signal can vary widely in both amplitude and phase depending on the intensity and relative propagation time of the multi-paths as well as the signal's bandwidth. The physical factors that influence small-scale fading include the multi-path richness of the propagation environment, the speed of the mobile, the speed of the surrounding objects and the transmission bandwidth of the signal.
Using the power delay profile of the propagation channel, multi-path channel parameters can be extracted that quantify the time dispersive properties of the multi-path propagation channel. If the channel's delay spread is much greater than the symbol duration, several attenuated and time delayed replicas of the transmitted waveform arrive at the receiver. When this occurs, the channel is said to exhibit frequency selective fading, as different spectral components experience different attenuation levels. On the other hand, if the delay spread of the channel is smaller than the symbol period, all spectral components experience the same channel attenuation level and the received signal is said to undergo a frequency flat fading. In this type of fading the spectral characteristics of the transmitted signal are preserved at the receiver, except that the received signal's strength changes with time as a result of the fluctuation in the channel's gain due to multipath fading.
For mobile radio applications, the channel is time variant because of the relative motion between a transmitter and a receiver or by movement of objects within the channel, which causes variations in the received signal's amplitude and phase. Consider a mobile receiver moving at a constant velocity v from point A to point B, such that the distance moved by the mobile receiver is d. The difference between the distance travelled by the waves from the source to A and B isΔl=vΔt cos θ,  (1)where Δt is the time required for the mobile to travel from A to B when traveling at a fixed velocity v, and θ is the angle of arrival in the azimuth plane for each ray that is assumed to remain constant. The phase change in the received signal due to the difference in the path lengths is therefore
                                          Δ            φ                    =                                                    2                ⁢                πΔ                ⁢                                                                  ⁢                l                            λ                        =                                          2                ⁢                πΔ                ⁢                                                                  ⁢                t                ⁢                                                                  ⁢                cos                ⁢                                                                  ⁢                θ                            λ                                      ,                            (        2        )            where λ is the carrier wavelength. The rate of change in Δφ with respect to time is the apparent change in frequency or Doppler shift and is given by
                                          f            D                    =                                                    1                                  2                  ⁢                  π                                            ⁢                              Δφ                                  Δ                  ⁢                                                                          ⁢                  t                                                      =                                                            v                  λ                                ⁢                cos                ⁢                                                                  ⁢                θ                            =                                                                    vf                    C                                    c                                ⁢                cos                ⁢                                                                  ⁢                θ                                                    ,                            (        3        )            where fc is the carrier frequency and c denotes the speed of light in a vacuum (c=3×108 m/s). The maximum value of the Doppler shift is vfc/c.
Therefore, if the mobile receiver is moving in the direction of arrival of the wave, the Doppler shift is positive, and if the mobile receiver is moving away from the direction of arrival of the wave, the Doppler shift is negative. It should be noted that the Doppler shift applies to a single multi-path. Each multi-path is subject to a specific Doppler shift, which can be positive or negative depending on the direction of motion compared to the azimuth angle of arrival of the particular multi-path. For a given direction of receiver motion, the range of different Doppler shifts arising due to multi-paths arriving from different azimuth directions results in a Doppler frequency spread calculated as the difference between the largest positive and largest negative shift among the multi-paths.